Supremum & Infimum
0.1 Bounded Sets (Real Numbers)
A set \(A \subset \mathbb{R}\) is bounded from above if there exists \(M \in \mathbb{R}\), called an upper bound of \(A\), such that \(x \leq M\) for all \(x \in A\). Similarly, \(A\) is bounded from below if there exists \(m \in \mathbb{R}\), called a lower bound of \(A\), such that \(x \geq m\) for all \(x \in A\). A set \(A\) is bounded if it is bounded both from above and below, equivalently if \(A \subset [m,M]\) for some \(m,M\in\mathbb{R}\).
0.2 Supremum & Infimum (Real Numbers)
The supremum and infimum are key concepts for defining completeness. In real analysis, we can use the [[order|ordering properties]] of \(\mathbb{R}\) to define the supremum and infimum of a set of real numbers.
The set of real numbers \(A\subset \mathbb{R}\) has an upper bound \(M \in \mathbb{R}\) such that \(M \leq M'\) for every upper bound \(M'\) of \(A\), then \(M\) is the least upper bound or supremum of \(A\), denoted \[ M=\sup{A}. \] Similarly, if \(A \subset \mathbb{R}\) has a lower bound \(m \in \mathbb{R}\) such that \(m\geq m'\) for every lower bound \(m'\) of \(A\), then \(m\) is the greatest lower bound or infimum of \(A\), denoted \[ m=\inf{A}. \]
If \(A=\{ x_{i}:i \in I \}\) is an indexed subset of \(\mathbb{R}\) we write \[ \sup{A}=\sup_{i \in I}{x_{i}},\quad \quad \inf{A}=\inf_{i \in I}{x_{i}}. \] Note that if \(M,m \in A\) then the supremum \(M\) and infimum \(m\) are equivalent to the maximum and minimum respectively. The key point is that the supremum and infimum need not belong to the set \(A\).
If a set \(A \subset \mathbb{R}\) is not bounded from above, then \(\sup_{}{A}= \infty\) and if \(A\) is not bounded from below then \(\inf_{}{A}=-\infty\).
0.3 Supremum & Infimum Properties
If \(A \subset \mathbb{R}\) then \(M = \sup_{}{A}\) if and only if: 1. \(M\) is an upper bound of \(A\); 2. \(\forall M'<M,~\exists x \in A~:~x>M'\).
Similarly, \(m = \inf_{}{A}\) if and only if: 1. \(m\) is a lower bound of \(A\); and 2. \(\forall m'>m,~\exists x \in A~:~x<m'\).
We frequently use this proposition as follows:
- If \(M\) is an upper bound of \(A\); then \(\sup_{}{A}\leq M\).
- If you \(A\) is nonempty and bounded from above, then for every \(\varepsilon>0,~\exists x \in A~s.t. x>\sup_{}{A}-\varepsilon\).
0.4 Supremum & Infimum of Extended Real Sequences
Consider an extended real sequence \(\{ x_{n} \}_{n \in \mathbb{N}}\subseteq\overline{\mathbb{R}}=\mathbb{R}\cup \{ \pm \infty \}\). A number \(u \in \overline{\mathbb{R}}\) is an upper bound of the sequence if \(x_{n}\leq u\) for all \(n\), and the sequence is bounded above. Similarly, \(l \in \overline{\mathbb{R}}\) is a lower bound if \(x_{n}\geq l\) for all \(n\), and the sequence is bounded below. A sequence is bounded if it is both bounded above and below.
The supremum \(\sup_{n}{x_{n}}\) is the unique least upper bound of the sequence: it is an upper bound, and \(\sup_{n}x_{n}\leq u\) for every upper bound \(u\). The infimum \(\inf_{n}{x_{n}}\) is the unique greatest lower bound: it is a lower bound, and \(\inf_{n}x_{n}\geq l\) for every lower bound \(l\).
These are fundamental definitions in [[measure-theory-notes]] and Real Analysis, where they form the basis for defining limits, continuity and integrals, and in particular Limit Supremum & Limit Infimum.
For an extended real sequence \(\{ x_{n} \}\), \[ \inf_{n}{x_{n}}\leq x_{k}\leq \sup_{n}{x_{n}}\quad\text{for all }k. \]
Proof. The infimum is a lower bound and the supremum is an upper bound, so \(\inf_nx_n\leq x_k\leq\sup_nx_n\) for each \(k\) by definition.
If \(x_{n}\leq y_{n}\) for all \(n\) then \[ \inf_{n}{x_{n}}\leq \inf_{n}{y_{n}},\qquad \sup_{n}{x_{n}}\leq \sup_{n}{y_{n}}. \]
Proof. For every \(n\), \(x_n\leq y_n\leq\sup_ky_k\), so \(\sup_ky_k\) is an upper bound of \(\{x_n\}\) and hence \(\sup_nx_n\leq\sup_ky_k\). Similarly \(\inf_kx_k\leq x_n\leq y_n\) for every \(n\), so \(\inf_kx_k\) is a lower bound of \(\{y_n\}\) and hence \(\inf_kx_k\leq\inf_ny_n\).
For sequences \(\{ x_{n} \},\{ y_{n} \}\) (with no \(\infty-\infty\)), \[ \sup_{n}(x_{n}+y_{n})\leq \sup_{n}{x_{n}}+\sup_{n}{y_{n}},\qquad \inf_{n}{(x_{n}+y_{n})}\geq \inf_{n}{x_{n}}+\inf_{n}{y_{n}}. \]
Proof. For every \(n\), \(x_n+y_n\leq\sup_kx_k+\sup_ky_k\), so the right-hand side is an upper bound of \(\{x_n+y_n\}\) and hence dominates its supremum. The infimum statement is analogous. (We cannot argue with an index \(m\) attaining the supremum, as the supremum need not be attained.)
For nonnegative sequences \(\{ x_{n} \}, \{ y_{n} \}\), \[ \sup_{n}{(x_{n}y_{n})}\leq \sup_{n}{x_{n}}\cdot \sup_{n}{y_{n}},\qquad \inf_{n}{(x_{n}y_{n})}\geq \inf_{n}{x_{n}}\cdot \inf_{n}{y_{n}}. \]
Proof. As for sums, using \(0\leq x_n\leq\sup_kx_k\) and \(0\leq y_n\leq\sup_ky_k\) to bound \(x_ny_n\), and similarly with the infima. Nonnegativity is needed: for \(x_n=y_n=-n\) we have \(\sup x_n=\sup y_n=-1\), but \(\sup_n x_ny_n=\sup_n n^2=+\infty>1\).